It's a funny situation at the moment because most attempts are just sort of shoehorning LLM chat into a game that doesn't really need it, and then gamers quite rightly mock the results which don't really add anything to the game.
I think there's a bit of a lack of knowledge of game history on both sides, because we already know it can be fun when a game is designed around it. It worked in Façade (2005!) and it worked in Starship Titanic (1998!!), and both of those could have worked even better with the technology we have today.
One type of game that seems like an obvious one to me, is something a bit like The Sims except that the characters' decisions and speech are LLM-controlled. Maybe one of the characters is player-controlled, so you can interact and talk with everyone else. Or to describe it from the other direction, it'd be a kind of interactive fiction like AI Dungeon invented, but grounded in a physically simulated setting.
I sometimes wonder if Façade actually used neural networks for text generation or just simple markov chains... I remember pewdiepie saying melon and getting kicked out lol. Also the Sims would be awesome but imagine an RPG with rich worldbulding, and LLM's interacting with the lore, like in an Elder Scrolls or Fallout title. Wouldn't it be mind blowing?